Optimization with Non-Differentiable Constraints with Applications to Fairness, Recall, Churn, and Other Goals

Abstract

We show that many machine learning goals, such as improved fairness metrics, can be expressed as constraints on the model's predictions, which we call rate constraints. We study the problem of training non-convex models subject to these rate constraints (or any non-convex and non-differentiable constraints). In the non-convex setting, the standard approach of Lagrange multipliers may fail. Furthermore, if the constraints are non-differentiable, then one cannot optimize the Lagrangian with gradient-based methods. To solve these issues, we introduce the proxy-Lagrangian formulation. This new formulation leads to an algorithm that produces a stochastic classifier by playing a two-player non-zero-sum game solving for what we call a semi-coarse correlated equilibrium, which in turn corresponds to an approximately optimal and feasible solution to the constrained optimization problem. We then give a procedure which shrinks the randomized solution down to one that is a mixture of at most m + 1 deterministic solutions, given m constraints. This culminates in algorithms that can solve non-convex constrained optimization problems with possibly non-differentiable and non-convex constraints with theoretical guarantees. We provide extensive experimental results enforcing a wide range of policy goals including different fairness metrics, and other goals on accuracy, coverage, recall, and churn.

Optimizing Nondecomposable Data Dependent Regularizers via Lagrangian Reparameterization offers Significant Performance and Efficiency Gains

“…Interestingly, this reparameterization naturally leads to a Lagrangian based procedure where existing SGD based methods can be employed with little to no change, see Figure 2. While recent results suggest that optimizing nondecomposable data-dependent regularizers may be challenging Cotter et al (2018), our development shows that a sizable subclass of such regularizers indeed admit simple solution schemes. Our overall procedure comes with convergence rate guarantees and optimal per-iteration complexity.…”

“…Moreover, Column 2 compares Alg. 3 with classical dual ascent procedures from Cotter et al (2018). Here, full projections refers to computing Π C (•) after every inner iteration in Alg.…”

“…Essentially, these applications suggest that the performane of a model in expectation (on the entire population), does not automatically guarantee that the model will perform well on specific subgroups. Motivated by these issues encountered in various real world problems, recently Cotter et al (2018) presents a comprehensive study of the computational aspects of learning problems with rate constraints -there, the constrained setting is preferred (over penalized setting) due to several reasons including its ease of use for a practitioner. The authors show that for a general class of constraints, a proxy-Lagrangian based method must be used because the Lagrangian is not optimal.…”

Tuning Fairness by Balancing Target Labels

“…In Goh et al [14] and Zafar et al [33,34], suitable majorization-minimization/convexconcave procedures [24] were derived. Furthermore, such constrained optimization approaches may lead to more unstable training, and often yield classifiers with both worse accuracy and more unfair [7]. Our proposed method can be solved by off-the-shelf packages, for example, we can use GPC packages by Dezfouli and Bonilla [8] or Gardner et al [13], which only need conditional likelihood evaluation as a black-box function.…”

Group-based Fair Learning Leads to Counter-intuitive Predictions

“…Other than post-processing, constrained optimization provides another common approach, in which enforcing group parity is framed as a constraint on a minimum loss objective. The resulting constrained optimization problem is then solved through the use of Lagrange multipliers [22,12,9,5,6,1,16].…”